Chaos, Communication and Consciousness Module PH19510 Lecture 16 Chaos.

Chaos, Communication and ConsciousnessModule PH19510

Lecture 16

Chaos

Overview of Lecture

The deterministic universe What is Chaos ? Examples of chaos Phase space Strange attractors Logistic differences – chaos in 1D Instability in the solar system

Chaos – Making a New Science

James Gleick Vintage ISBN

0-749-38606-1

£8.99 http://www.around.com

Before Chaos

A Newtonian Universe : Fully deterministic with complete predictability

of the universe. Laplace thought that it would be possible

to predict the future if we only knew the right equations. "Laplace's Demon."

Causal Determinism

Weather Control in a deterministic universe von Neumann (1946)

Identify ‘critical points’ in weather patterns using computer modelling

Modify weather by interventions at these points

Use as weapon to defeat communism

Modern Physics and the Deterministic Universe Relativity (Einstein)

Velocity of light constantLength and Time depend on observer

Quantum TheoryLimits to measurementTruly random processes

Chaos

What is Chaos ?

Observed in non-linear dynamic systems Linear systems

variables related by linear equations equations solvable behaviour predictable over time

Non-Linear systems variables related by non-linear equations equations not always solvable behaviour not always predictable

What is Chaos ?

Not randomness Chaos is

deterministic – follows basic rule or equationextremely sensitive to initial conditionsmakes long term predictions useless

Examples of Chaotic Behaviour

Dripping Tap Weather patterns Population Turbulence in liquid or gas flow Stock & commodity markets Movement of Jupiter's red spot Biology – many systems Chemical reactions Rhythms of heart or brain waves

Phase Space

Mathematical map of all possibilities in a system

Eg Simple Pendulum Plot x vs dx/dt Damped Pendulum

Point Attractor Undamped Pendulum

Limit cycle attractor

Damped Pendulum – Point Attractor

velo

cit

y

position

Undamped Pendulum – Limit Cycle Attractor

The ‘Strange’ Attractor

Edward Lorentz From study of

weather patterns Simulation of

convection in 3D Simple as possible

with non-linear terms left in. The Lorenz Attractor

bzxydt

dz ,xzyrx

dt

dy ,xy

dt

dx

Sensitivity to initial conditions

Blue & Yellow differ in starting positions by 1 part in 10-5

Evolution of system in phase space

Simplest Chaotic System

Logistic equations Model populations in biological system

tt1t x1x kx

What happens as we change k ?

k<3 – Fixed Point Attractor

At low values of k (<3), the value of xt eventually stabilises to a single value - a fixed point attractor

k=3 – Limit Cycle Attractor

When k is 3, the system changes to oscillate between two values.

This is called a bifurcation event.

Now have a limit cycle attractor of period 2.

k=3.5 – 2nd Bifurcation event

When k is 3.5, the system changes to oscillate between four values.

Now have a limit cycle attractor of period 4.

k=3.5699456 – Onset of chaos

When k is > 3.5699456 x becomes chaotic

Now have a Aperiodic Attractor

Onset of chaos

Feigenbaum diagram

Shows bifurcation branches

Regions of order re-appear

Figure is ‘scale invariant’ k

xt

k = 3.5699456 Onset of chaos

Instability in the Solar System

3 Body ProblemPossible to get exact, analytical solution for 2

bodies (planet+satellite)No exact solution for 3 body systemPossible to arrive at approximation by making

assumptionsSolutions show chaotic motion

The moon cannot have satellites

Asteroid Orbits

Jupiter

Mars

Asteroid Orbits

The Kirkwood gap

Daniel Kirkwood (1867) No asteroids at 2.5 or 3.3 a.u. from sun 2:1 & 3:1 resonance with Jupiter Jack Wisdom (1981) solved three-body problem

of Jupiter, the Sun and one asteroid at 3:1 resonance with Jupiter.

Showed that asteroids with such specifications will behave chaotically, and may undergo large and unpredictable changes in their orbits.

Review of Lecture

The deterministic universe What is Chaos ? Examples of chaos Phase space Strange attractors Logistic differences – chaos in 1D Instability in the solar system